Conditional Stability and Instability Estimates for Inverse Random Source Problems

This thesis is motivated by the study of passive imaging problems. Such problems can be characterised as scattering problems under uncontrolled incident waves. We focus specifically on inverse random source problems, a subclass of passive imaging problems. These are linear inverse problems where we aim to recover the random source from measurements of the scattered field on some surface enclosing the source region. In general, such problems lack uniqueness because of the existence of non-radiating sources. However, for centred Gaussian source processes, it can be shown that such inverse problems are uniquely solvable. Building on this uniqueness, we investigate the stability of these inverse problems. In particular, we prove that such problems satisfy Hölder-logarithmic stability estimates under Sobolev-type smoothness assumptions on the source power, depending on the wave number. For a fixed wave number, we also establish logarithmic instability estimates that only slightly differ in the order of the exponential ill-posedness compared to the conditional stability estimates. Furthermore, the used methodology enables us to derive convergence rates for the inverse problems for a variety of spectral regularisation methods under suitable parameter choice rules. The first part of the thesis introduces conditional stability estimates within a generalised framework. In particular, we introduce a notion of local conditional stability, tailored to fit into the localised concept of variational source conditions. In the second part we focus on random processes in Hilbert spaces that serve as models for the random sources in the passive imaging. The third part presents the main contributions of this thesis, namely the conditional stability and instability estimates for the inverse random source problems. We study three different problems, and for each we use complex geometric optics solutions to establish variational source conditions, leading to Hölder-logarithmic stability and convergence rates. This part concludes with establishing logarithmic instability for each of the problems. Both logarithmic stability and instability rely on natural Sobolev-type smoothness assumptions on the source power of uncorrelated centred Gaussian source processes. Finally, we numerically investigate the convergence rates.